Analytic Geometry
Defining Trigonometric Ratios
Name ___________________
Date __________ Per _____
MCC9-12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Pearson: TL 7-3
Warm-up:
Contractors are repairing and painting the tower on the county courthouse. They need to calculate the height of the tower
to know what equipment they will need. The diagram below shows how they measured the height of the tower using a
laser light, a post, and a tape measure. They put the post in the ground 3 feet in front of a 3-foot-tall tripod that has a laser
on it. The height of the post is 8 feet, and the post is perpendicular to the ground. The laser is aimed just above the post
so that the light beam hits the top of the tower. The post is 45 feet from the base of the courthouse tower.
1. How many triangles are shown in the diagram?
2. Which angle do you the triangle share in common?
3. What is the height of the tower?
Key Concepts:
____________________ is the study of triangles and the relationships between their sides and
the angles between these sides. In this lesson, we will learn about the ratios between angles and side
lengths in right triangles.
The acute angle that is being used for the ratio is known as the reference angle. It is
commonly marked with the symbol _____ (theta).
The length of the hypotenuse remains the same,
but the sides that are _______________ or
_______________ for each acute angle will be
different for different reference angles. The two rays of each acute angle in a right
triangle are made up of a _____ and the _______________. The leg is called the
____________________ (_____) to the angle. The side of the triangle opposite the
reference angle is called the ____________________ (_____).
A ratio of the lengths of two sides of a right triangle is called a _________________________.
The three most common ratios are _____________, ______________, and _____________.
You must always remember to check your calculator. It needs to be in __________ mode in order
to calculate the answers correctly.
Let’s make sure we can use our calculator:
Sin 48 = _______
Tan 22 = _______
Cos 43 = _______
To find the angle measure given the decimal equivalent: (use sin-1, cos-1, tan-1 on the calculator)
Sin _____ = .2835
Cos _____ = .08375
Tan _____ = 1.25
Finding Angle Measure using Right Triangle Trig
Example 1: Given the triangle below, find the following (ratio and decimal form):
sin A ___________
cos A _________
tan A __________
In these instances you are given or can find the lengths of the sides. You would only have to use a
calculator to convert ratios to decimals. You would not be using the sine, cosine, or tangent keys on
your calculator.
For the following problems, you will use the sine, cosine, and tangent keys on your calculator to find
the ratios in decimal form. You are given angles in degrees. Make sure that your calculator is in
degree mode.
Example 2: Use a scientific calculator to determine the values accurate to the nearest ten
thousandths.
1. sin 37 ___________
2. cos 29 _________
3. tan8 __________
4. sin 67 ___________
5. tan 79 _________
6. cos33 __________
You can use the lengths of the sides of a triangle or given ratios to find the measure of angles in right
1
triangles. Now you will use the 2nd sine, cosine, or tangent called the inverse sine ( sin ), cosine ( cos
1
), or tangent ( tan ). Remember, your 2nd key and trig functions will only be used to find angle
measures.
Example 3: Find the measure of each angle to the nearest degree.
1. sin A  .5 _________
2. cos B  .6 _______
3. tan C  .5773 ________
4. sin D  .7071_________
5. tan E  1.7321 _______
6. cos F  .3090 ________
Example 4: Find the measure of the angles in the given triangle. mA ________, mC ______
Example 5: Set up the trigonometric relationship
and use a calculator to determine x.
.
1
Example 6: Set up the trigonometric relationship and use a calculator to determine
the missing value.
Cofunctions: Sine and Cosine as Complementary Angles
sin A 
sin B 
cos B 
cos A 
In, ABC sin A  ________ and sin B  ________
The two acute angles in a right triangle have a sum of 90 . They are _______________ angles. If one
acute angle has a measure of x, the other angle has a measure of 90  _____ .
The sine and cosine cofunctions can be written as: sin   cos 90   
cos   sin  90   
Example 7: Find sin 28 if cos62  .469
Set up the identity: sin   cos  90   
Substitute the values of the angles and simplify: sin 28  cos  90  28
sin 28  cos62
Verify the identity by checking with your calculator.
Example 8: Complete the table below using sine and cosine identities.
Example 9: Find a value of  for which sin   cos15 is true.
Example 10: Complete the table below using the sine and cosine identities.
Homework
Name______________________
Similar Triangles and Right Triangle Trig
Find the value of each trigonometric ratio.
1.
4.
2.
3.
5.
6.
Write the trigonometric equation you would use to solve for x in the following triangles.
7.
8.
9.
10.
11.
12.
13. Solve the right triangle. Find all the missing sides and angles.
Analytic Geometry
Practice with Trig Functions
Name ______________________
Date ________
Period _____
Find the following using a calculator and rounding to four decimal places.
(Make sure your calculator is in degree mode).
1. tan 48º
2. sin 89º
3. cos 14º
4. tan 73º
5. cos 48º
6. tan 89º
7. sin 14º
8. sin 73º
Find x.
9. sin34 
x
20
10. tan10 
49
x
11. cos 72 
x
14
8
x
13. tan 55 
x
7
14. sin 29 
13
x
12. cos 48 
Find angle A.
15. tan A 
8
12
16. cos A 
12
15
17. sin A 
24
25
18. cos A 
8
12
19. sin A 
5
3
20. tan A 
24
7
Answers
x
17
37
12) tan x 
46
7)
8)
cos 71 
13) f = 9.3338
x
17
9)
e = 4.8069
tan 72 
18
x
10)
sin x 
39
26
11) cos x 
47
39
D  59
1)1.1106 2) .9999 3) .9703 4) 3.2709 5) .6691 6) 57.2810 7) .2419 8) .9563
12) 11.9558 13) 9.9970
cos 58 
9) 11.1839 10) 277.8928 11) 4.3262
14) 26.8146 15) 33.6901 16) 36.8699 17) 73.7398 18) 48.1897 19) NP 20) 73.7398